9 mark for review let f be a function such that lim f(x) = ∞. which of the following statements must be…

Explanation:

Step1: Recall limit and asymptote definitions

The notation $\lim_{x\rightarrow a}f(x)=\infty$ means that as $x$ approaches $a$, the values of $f(x)$ increase without bound. A vertical - asymptote of the graph of $y = f(x)$ occurs at $x=a$ if either $\lim_{x\rightarrow a^{-}}f(x)=\pm\infty$, $\lim_{x\rightarrow a^{+}}f(x)=\pm\infty$, or $\lim_{x\rightarrow a}f(x)=\pm\infty$.

Step2: Analyze option A

$\lim_{x\rightarrow 5}f(x)=\infty$ does not imply $\lim_{x\rightarrow 5^{-}}f(x)=\infty$. The one - sided limit $\lim_{x\rightarrow 5^{-}}f(x)$ could be $-\infty$ while $\lim_{x\rightarrow 5^{+}}f(x)=\infty$ and still $\lim_{x\rightarrow 5}f(x)=\infty$.

Step3: Analyze option B

The fact that $\lim_{x\rightarrow 5}f(x)=\infty$ does not mean that $f(x)$ is undefined at $x = 5$. A function can be defined at a point and still have an infinite limit as $x$ approaches that point. For example, $f(x)=\frac{1}{(x - 5)^2}+1$ is defined at $x = 5$ ($f(5)=1$) but $\lim_{x\rightarrow 5}f(x)=\infty$.

Step4: Analyze option C and D

If $\lim_{x\rightarrow 5}f(x)=\infty$, by the definition of a vertical asymptote, the graph of $y = f(x)$ has a vertical asymptote at $x = 5$. Options C and D are the same in the provided image.

Answer:

C. The graph of $f$ has a vertical asymptote at $x = 5$